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Theorem distgp 22635
Description: Any group equipped with the discrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
distgp.1 𝐵 = (Base‘𝐺)
distgp.2 𝐽 = (TopOpen‘𝐺)
Assertion
Ref Expression
distgp ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp)

Proof of Theorem distgp
StepHypRef Expression
1 simpl 483 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ Grp)
2 simpr 485 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 = 𝒫 𝐵)
3 distgp.1 . . . . . 6 𝐵 = (Base‘𝐺)
43fvexi 6677 . . . . 5 𝐵 ∈ V
5 distopon 21533 . . . . 5 (𝐵 ∈ V → 𝒫 𝐵 ∈ (TopOn‘𝐵))
64, 5ax-mp 5 . . . 4 𝒫 𝐵 ∈ (TopOn‘𝐵)
72, 6syl6eqel 2918 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 ∈ (TopOn‘𝐵))
8 distgp.2 . . . 4 𝐽 = (TopOpen‘𝐺)
93, 8istps 21470 . . 3 (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵))
107, 9sylibr 235 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopSp)
11 eqid 2818 . . . . . 6 (-g𝐺) = (-g𝐺)
123, 11grpsubf 18116 . . . . 5 (𝐺 ∈ Grp → (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
1312adantr 481 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
144, 4xpex 7465 . . . . 5 (𝐵 × 𝐵) ∈ V
154, 14elmap 8424 . . . 4 ((-g𝐺) ∈ (𝐵m (𝐵 × 𝐵)) ↔ (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
1613, 15sylibr 235 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g𝐺) ∈ (𝐵m (𝐵 × 𝐵)))
172, 2oveq12d 7163 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = (𝒫 𝐵 ×t 𝒫 𝐵))
18 txdis 22168 . . . . . . 7 ((𝐵 ∈ V ∧ 𝐵 ∈ V) → (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵))
194, 4, 18mp2an 688 . . . . . 6 (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵)
2017, 19syl6eq 2869 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = 𝒫 (𝐵 × 𝐵))
2120oveq1d 7160 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝒫 (𝐵 × 𝐵) Cn 𝐽))
22 cndis 21827 . . . . 5 (((𝐵 × 𝐵) ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵m (𝐵 × 𝐵)))
2314, 7, 22sylancr 587 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵m (𝐵 × 𝐵)))
2421, 23eqtrd 2853 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝐵m (𝐵 × 𝐵)))
2516, 24eleqtrrd 2913 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
268, 11istgp2 22627 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
271, 10, 25, 26syl3anbrc 1335 1 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  𝒫 cpw 4535   × cxp 5546  wf 6344  cfv 6348  (class class class)co 7145  m cmap 8395  Basecbs 16471  TopOpenctopn 16683  Grpcgrp 18041  -gcsg 18043  TopOnctopon 21446  TopSpctps 21468   Cn ccn 21760   ×t ctx 22096  TopGrpctgp 22607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397  df-0g 16703  df-topgen 16705  df-plusf 17839  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045  df-sbg 18046  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-cn 21763  df-cnp 21764  df-tx 22098  df-tmd 22608  df-tgp 22609
This theorem is referenced by: (None)
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